Mathematics of Computation

Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations

Author(s): G. N. Milstein; M. V. Tretyakov.
Journal: Math. Comp. 69 (2000), 237-267.
MSC (1991): Primary 35K55, 60H10, 60H30, 65M99
Posted: May 21, 1999
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

1.: M. Bossy, D. Talay. A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comp. 66 (1997), pp. 237-267. MR 97c:60233
2.: V.G. Danilov, V.P. Maslov, K.A. Volosov. Mathematical Modelling of Heat and Mass Transfer Processes. Kluwer Academic Publishers, Dordrecht, 1995 (Engl. transl. from Russian 1987). MR 98a:35013
3.: K. Eriksson, C. Johnson. Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995), pp. 1729-1749. MR 96i:65081
4.: C.A.J. Fletcher. Computational Techniques for Fluid Dynamics. Volumes I, II, Springer, 1991. MR 92e:76042; MR 92f:76065
5.: M.I. Freidlin. Functional Integration and Partial Differential Equations. Princeton Univ. Press, Princeton, 1985. MR 87g:60066
6.: M.I. Freidlin. Markov Processes and Differential Equations: Asymptotic Problems. Birkhäuser, Basel, 1996. MR 97f:60150
7.: I.M. Gelfand. Some problems in the theory of quasi-linear equations. Uspehi Mat. Nauk 14 (1959), pp. 237-267. (Russian) MR 22:1736 (English transl., Amer. Math. Soc. Trans. (2) 29 (1963), 295-381. MR 27:3921)
8.: A.M. Il'in. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Nauka, Moscow, 1989. (Russian) MR 90i:65032 (English transl., Amer. Math. Soc., Providence, RI, 1992. MR 93g:35016)
9.: A.M. Il'in, O.A. Oleinik. Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of time. Mat. Sbornik 51 (1960), pp. 237-267. (Russian) MR 22:11222
10.: E. Hairer, S.P. Nørsett, G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 1993. MR 94c:65005
11.: J. Kevorkian, J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer, 1996. MR 97k:34001
12.: P.E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, 1992. MR 94b:60069
13.: O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'ceva. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, R.I., 1988 (Engl. transl. from Russian 1967). MR 39:3159
14.: G.N. Milstein. Numerical Integration of Stochastic Differential Equations. Kluwer Academic Publishers, Norwell, MA, 1995 (engl. transl. from Russian 1988). MR 96e:65003
15.: G.N. Milstein. Solving the first boundary value problem of parabolic type by numerical integration of stochastic differential equations. Theory Prob. Appl. 40 (1995), pp. 237-267. MR 97f:35087
16.: G.N. Milstein. Application of numerical integration of stochastic equations for solving boundary value problems with the Neumann boundary conditions. Theory Prob. Appl. 41 (1996), pp. 237-267. MR 97j:60107
17.: G.N. Milstein. Weak approximation of a diffusion process in a bounded domain. Stochastics and Stochastics Reports 62 (1997), pp. 237-267. MR 98h:60116
18.: G.N. Milstein. The probability approach to numerical solution of nonlinear parabolic equations. Preprint No. 380, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, 1998 (submitted).
19.: G.N. Milstein, M.V. Tretyakov. Numerical methods in the weak sense for stochastic differential equations with small noise. SIAM J. Numer. Anal., 34 (1997), pp. 237-267. MR 98i:60055
20.: G.N. Milstein, M.V. Tretyakov. Numerical methods for nonlinear parabolic equations with small parameter based on probability approach. Preprint No. 396, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, 1998.
21.: J. von Neumann, R. Richtmyer. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21 (1950), pp. 237-267. MR 12:289b
22.: E. Pardoux, D. Talay. Discretization and simulation of stochastic differential equations. Acta Appl. Math. 3 (1985), pp. 237-267. MR 86h:60114
23.: A. Quarteroni, A. Valli. Numerical Approximation of Partial Differential Equations. Springer, 1994. MR 95i:65005
24.: R.D. Richtmyer, K.W. Morton. Difference Methods for Initial-Value Problems. Interscience, New York, 1967. MR 36:3515
25.: P.J. Roache. Computational Fluid Dynamics. Hermosa, Albuquerque, N.M., 1976. MR 53:15094
26.: H.-G. Roos, M. Stynes, L. Tobiska. Numerical Methods for Singularity Perturbed Differential Equations: Convection-Diffusion and Flow Problems. Springer, 1996. MR 99a:65134
27.: A.A. Samarskii. Theory of Difference Schemes. Nauka, Moscow, 1977. (Russian) MR 58:3288 (German transl., Akademische Verlag. Geest & Portig, Leipzig, 1984. MR 86e:65003)
28.: J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer, 1983. MR 84d:35002
29.: M.E. Taylor. Partial Differential Equations III: Nonlinear Equations. Springer, 1996; corrected reprint, 1997. MR 98k:35001
30.: E.V. Vorozhtsov, N.N. Yanenko. Methods for the Localization of Singularities in Numerical Solutions of Gas Dynamic Problems. Springer, 1990 (Engl. transl. from Russian 1985). MR 91m:76069

Retrieve articles in Mathematics of Computation with MSC (1991): 35K55, 60H10, 60H30, 65M99

G. N. Milstein
Affiliation: Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-10117 Berlin, Germany
Email: milstein@wias-berlin.de

M. V. Tretyakov
Affiliation: Department of Mathematics, Ural State University, Lenin str. 51, 620083 Ekaterinburg, Russia
Email: Michael.Tretyakov@usu.ru

DOI: 10.1090/S0025-5718-99-01134-5
PII: S 0025-5718(99)01134-5
Keywords: Semilinear parabolic equations, reaction-diffusion systems, probabilistic representations for equations of mathematical physics, stochastic differential equations with small noise
Received by editor(s): April 7, 1998
Posted: May 21, 1999
Copyright of article: Copyright 1999, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS